Elliptic functions and equations of modular curves

Borisov, Lev A.; Gunnells, Paul E.; Popescu, Sorin

doi:10.1007/s002080100238

Cited by 8 publications

(17 citation statements)

References 15 publications

(27 reference statements)

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“…In addition, these s i satisfy the equation 7(s 2 1 + s 2 2 + s 2 3 ) = 5(s 1 + s 2 − s 3 ) 2 , which establishes an isomorphism of X 1 (7) with a conic in P 2 (C). For more details see [3]. Proposition 6.3.…”

Section: Relation Amongmentioning

confidence: 98%

“…The extension K/Q is Galois, with Gal(K/Q) ≃ Z/3Z, and a generator of the Galois group maps w to w 2 − 2. The norm and trace of an element r = a + bw + cw 2 are given respectively by N(r) = a 3 − a 2 b + 5a 2 b − 2ab 2 − abc + 6ac 2 + b 3 − b 2 c − 2bc 2 + c 3 and Tr(r) = 3a − b + 5c. We fix an isomorphism…”

Section: The Cubic Field Of Discriminant 49mentioning

confidence: 99%

“…Remark 5.7. We can represent the modular forms for Γ(p) as power series in variables q 1 , q 2 , q 3 .…”

Section: Eisenstein Seriesmentioning

confidence: 99%

“…It is easy to check that the three quadrics in the statement generate this subspace. Alternatively, one can look at the eight translates of the cusp (i∞) 3 given by (F 0 : F 1 : F 2 : F 4 ) = (1 : 0 : 0 : 0) by the action of SL(2, F 7 ) and look at the quadrics that vanish at these points. Proposition 8.2.…”

Section: The Symmetric Hilbert Modular Threefoldmentioning

confidence: 99%

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On Hilbert modular threefolds of discriminant 49

Borisov

Gunnells

2012

Sel. Math. New Ser.

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To Don Zagier, on the occasion of his 60th birthday.Abstract. Let K be the totally real cubic field of discriminant 49, let O be its ring of integers, and let p ⊂ O be the prime over 7. Let Γ(p) ⊂ Γ = SL 2 (O) be the principal congruence subgroup of level p. This paper investigates the geometry of the Hilbert modular threefold attached to Γ(p) and some related varieties. In particular, we discover an octic in P 3 with 84 isolated singular points of type A 2 .

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Section: Relation Amongmentioning

confidence: 98%

Section: The Cubic Field Of Discriminant 49mentioning

confidence: 99%

“…Remark 5.7. We can represent the modular forms for Γ(p) as power series in variables q 1 , q 2 , q 3 .…”

Section: Eisenstein Seriesmentioning

confidence: 99%

Section: The Symmetric Hilbert Modular Threefoldmentioning

confidence: 99%

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On Hilbert modular threefolds of discriminant 49

Borisov

Gunnells

2012

Sel. Math. New Ser.

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“…Furthermore, arithmetic aspects of the theory can be found in a well-known book of Shimura ( [13]). The uniformization of modular curves is used to compute equations of modular curves X 0 (N ) in [2,6,14,16].…”

Section: Introductionmentioning

confidence: 99%

Birational Maps of X(1) into P2

Mikoć¹,

Muić²

2013

Glas. Mat. Ser. III

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Abstract. In this paper we study birational maps of modular curve X(1) attached to SL 2 (Z) into the projective plain P 2 . We prove that every curve of genus 0 and degree q in P 2 can be uniformized by modular forms for SL 2 (Z) of weight 12q but not with modular forms of smaller weight, and that the corresponding uniformization can be chosen to be a birational equivalence. We study other regular maps X(1) −→ P 2 and we compute the equation of obtained projective curve. We provide numerical examples in SAGE.

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Toric varieties and modular forms

Borisov

Gunnells

2001

Invent. math.

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Let N ⊂ R r be a lattice, and let deg : N → C be a piecewiselinear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on deg, the data (N, deg) determines a function f : H → C that is a holomorphic modular form of weight r for the congruence subgroup Γ 1 (l). Moreover, by considering all possible pairs (N, deg), we obtain a natural subring T (l) of modular forms with respect to Γ 1 (l). We construct an explicit set of generators for T (l), and show that T (l) is stable under the action of the Hecke operators. Finally, we relate T (l) to the Hirzebruch elliptic genera that are modular with respect to Γ 1 (l).

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Elliptic functions and equations of modular curves

Abstract: Abstract. Let p ≥ 5 be a prime. We show that the space of weight one Eisenstein series defines an embedding into P (p−3)/2 of the modular curve X 1 (p) for the congruence group Γ 1 (p) that is scheme-theoretically cut out by explicit quadratic equations.

Cited by 8 publications

References 15 publications

On Hilbert modular threefolds of discriminant 49

On Hilbert modular threefolds of discriminant 49

Birational Maps of X(1) into P2

Toric varieties and modular forms

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